Drought Risk Analysis in the Eastern Cape Province of South Africa: the Copula Lens

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Article Drought Risk Analysis in the Eastern Cape Province of South Africa: The Copula Lens

Christina M. Botai 1,* , Joel O. Botai 1,2,3 , Abiodun M. Adeola 1,4 , Jaco P. de Wit 1, Katlego P. Ncongwane 1,5 and Nosipho N. Zwane 1

1 South African Weather Service, Private Bag X097, Pretoria 0001, South Arica; [email protected] (J.O.B.); [email protected] (A.M.A.); [email protected] (J.P.d.W.); [email protected] (K.P.N.); [email protected] (N.N.Z.) 2 Department of Geography, Geoinformatics and Meteorology, University of Pretoria, Private Bag X020, Hatﬁeld 0028, South Africa 3 Department of Information Technology, Central University of Technology, Bloemfontein 9301, South Africa 4 Institute for Sustainable Malaria Control, University of Pretoria, Pretoria 0002, South Africa 5 School of Geography and Environmental Science, University of KwaZulu-Natal, Durban 4041, South Africa * Correspondence: [email protected]; Tel.: +27-12-367-6269

Received: 28 May 2020; Accepted: 3 July 2020; Published: 8 July 2020

Abstract: This research study was carried out to investigate the characteristics of drought based on the joint distribution of two dependent variables, the duration and severity, in the Eastern Cape Province, South Africa. The drought variables were computed from the Standardized Precipitation Index for 6- and 12-month accumulation period (hereafter SPI-6 and SPI-12) time series calculated from the monthly rainfall data spanning the last five decades. In this context, the characteristics of climatological drought duration and severity were based on multivariate copula analysis. Five copula functions (from the Archimedean and Elliptical families) were selected and fitted to the drought duration and severity series in order to assess the dependency measure of the two variables. In addition, Joe and Gaussian copula functions were considered and fitted to the drought duration and severity to assess the joint return periods for the dual and cooperative cases. The results indicate that the dependency measure of drought duration and severity are best described by Tawn copula families. The dependence structure results suggest that the study area exhibited low probability of drought duration and high probability of drought severity. Furthermore, the multivariate return period for the dual case is found to be always longer across all the selected univariate return periods. Based on multivariate analysis, the study area (particularly Buffalo City, OR Tambo and Alfred Zoo regions) is determined to have higher/lower risks in terms of the conjunctive/cooperative multivariate drought risk (copula) probability index. The results of the present study could contribute towards policy and decision making through e.g., formulation of the forward-looking contingent plans for sustainable management of water resources and the consequent applications in the preparedness for and adaptation to the drought risks in the water-linked sectors of the economy.

Keywords: drought; dependence measure; Standardized Precipitation Index; copula; return periods

1. Introduction The synchronous anomalous climatic conditions that society experiences today did not return; it never went away. This is especially true because the theme about mankind’s struggle with climatic adversaries has always been interwoven in the world’s stock of e.g., documentary, religion and mythology literature. In this regard, drought conditions are the most profound and omnipresent

Water 2020, 12, 1938; doi:10.3390/w12071938 www.mdpi.com/journal/water Water 2020, 12, 1938 2 of 20 adverse weather phenomena that manifests as an extended period of deficit precipitation relative to normal [1,2]. As a corollary, drought exhibits a global footprint mimicking the spatial–temporal nature of variability of precipitation. From the temporal viewpoint, drought conditions manifesting as insufficient or delayed precipitation is classified as a meteorological drought [1], with the definition relating to the imbalance of water availability with in-consistence to persistent lower than average precipitation [2]. The condition that relates to insufficient soil moisture is classified as an agricultural drought, with the impacts manifested from crop failure, leading to a treat in food security [3]. On the other hand, a hydrological drought describes a condition whereby water availability in water reservoirs is significantly below normal levels over a specific period, a condition attributed to low precipitation coupled with increased evapotranspiration rates, thereby threatening water security [4]. In general, the inherent impacts of drought are naturally complex and often resonate across social, environmental, and economic sectors, including water, agriculture, energy, health, tourism, etc. [5]. There is compelling evidence that drought is the most damaging and pressing natural disaster with the highest frequency, the most critical disaster with significant socio-economic and ecological losses as well as the most commodious impact [6,7], causing significant damage in infrastructure and property as well as well loss of life in general [8]. There is, therefore, a need to monitor and evaluate drought events, from global to regional timescales, so to alleviate the impacts and for preparedness measures for future events. Drought indices are traditionally used for drought assessment, monitoring, and prediction. Such indices have been developed based on various datasets types e.g., precipitation only (the Standardized Precipitation Index (SPI); [9]), streamflow only (the Standardized Streamflow Index (SSI); [10]) as well as those derived from a combination of hydro-meteorological parameters, such as precipitation and temperature (the Standardized Precipitation and Evapotranspiration Index (SPEI); [11]), and precipitation, temperature and streamflow (Palmer Modified Drought Index (PDSI); [12]). Most of these indices have been evaluated and confirmed to be appropriate tools for monitoring, assessment (features such as the onset, the intensity, severity and frequency) as well as prediction of drought [13]. The SPI, particularly, has been extensively used for drought assessment, across the world, for instance in South Africa [13–16], in Bangladesh [17,18], India [19], and China [20]. Drought analysis and monitoring has also been conducted based on SPEI, for examples see studies by [13,21,22] and references therein. Previous research studies on drought over South Africa have alluded to the occurrence of widespread and persistent drying conditions in most parts of the country [13–16,23]. Five of the most economically important/driven provinces of South Africa are recently recovering from severe drought, which caused negative socio-economic impacts, including the disruption of water supply and agricultural activities. These conditions have resulted in a significantly high cost of living, particularly to the most vulnerable communities, including farmers that mostly depend on rain-fed agriculture. The projected increases in the frequency of climate-related extremes, including drought disasters [24,25], pose a significant threat to South Africa given the vulnerability of the country, e.g., more than 90% of the country is arid to semi-arid [26]. Well-deserved attention needs to be given to drought features such as the duration, frequency, and severity, including the potential increase of drought disaster in general. Furthermore, effective measures of dealing with future drought and its inherent impacts are needed, so as to help inform the design of programs to reduce, mitigate, and prepare for future drought related impacts. Multivariate distribution methods, based on copulas, are often used to evaluate and model drought features, such as the onset, duration, frequency, intensity, and severity [27,28]. For example, studies by [27] used two-dimensional copula functions to model the joint drought duration and severity distribution by fitting exponential and gamma distribution functions to the drought features, respectively. In [29], copulas were used to construct and investigate hydrological droughts of the Yellow River in northern China. A study in Iran used a copula-based distribution function to model the joint distribution of drought duration and severity for the period of 1954–2003 [30]. In addition, [31] applied trivariate copulas on the SPI series and characterized drought events under Water 2020, 12, 1938 3 of 20

El-Nino, La-Nina, and natural states. Similarly, using trivariate copulas, [32] evaluated the joint behaviour of drought characteristics (e.g., duration, severity, and intensity) under changing climate in Oregon’s upper Klamath River basin. Furthermore, [33] evaluated the influence of the tail shape of various copula functions on drought bivariate frequency analysis. Several copulas from Archimedean and meta-elliptical families were used by [34] to model four-dimensional joint distributions on the SPI and evaluate drought characteristics such as drought duration, severity and interval time. In South Africa, the SPI drought index has been used to study drought characteristics at different timescales [13,15,16,34–37] at national and regional level. Studies by [13,15] focused on drought characteristics in the Free State and North West as well as in the Western Cape provinces, respectively. Owing to the economic value of drought impacts in South Africa, the literature on drought monitoring and prediction from the viewpoint of drought indicators has continued to grow. A salient feature in all these studies is the quantification of drought conditions using univariate methods. Unlike multivariate-based risk assessment, the univariate quantification of droughts are not duly suited for estimating the true impacts of droughts. In general, the use of multivariate copula functions to study drought characteristics has been widely supported in the literature. Research studies on drought characteristics based on multivariate copula-based distribution approaches have been rarely reported in South Africa, although such methods offer an improved alternative approach to study the joint distribution of drought characteristics, such as duration, severity, frequency, and intensity, as well as the minimum SPI values within specific drought period. In this regard, the joint distribution analysis of the drought duration, severity, and frequency can provide essential information regarding the historical drought events and such information could be used in water resources management and planning. For this purpose, the current research study is focused on utilizing the multivariate copula-based functions to analyze characteristics of the joint distribution of drought duration and severity in the Eastern Cape province of South Africa. The characteristics of the joint distribution of drought duration and severity is derived from the SPI time series computed from monthly rainfall observational data. The results of this research work contribute towards drought monitoring for effective water resources management and planning. This paper is structured as follows: first, a description of the study area is provided in Section2, before the key materials and underlying methods for this research are introduced in Section3. A detailed description of the results is given in Section4 while the paper concludes with the discussion of the findings and summary of concluding remarks and future research recommendations.

2. Study Area The Eastern Cape province is one of the nine provinces and the second largest province in South Africa after the Northern Cape, with an area of close to 169,000 km2. The province is located on the south-eastern part of the country, bordering the Western Cape and Northern Cape provinces to the west and north-west, respectively. The Eastern Cape province also borders the Free State province and Lesotho towards the north, KwaZulu-Natal province towards the north-eastern region and the Indian Ocean to the south-eastern and south regions, see Figure1. The Eastern Cape’s coastal areas are characterized by subtropical climatic conditions which predominate in KwaZulu-Natal and the Mediterranean climate of the Western Cape. Long, hot summers and moderate winters are mostly experienced in the Karoo towards the west, while the Great Escarpment towards Lesotho and the Free State experience snow in winter. Rainfall in the province exhibits bimodal patterns, with a winter rainfall (or all year rainfall) zone in the west, and summer rainfall zone in the east. On average, the Eastern Cape Province rainfall varies between 100 mm and 520 mm per year. Varying rainfall seasons result in varying growing periods, e.g., the summer seasonality advocates for the 4-Carbon (C4) grass production in the north, east and along the coastal belt, dominating with cattle and sheep production, whereas, the 3-Carbon (C3) grasses and shrubs predominate in the semi-arid central and western regions, favoring sheep and goat production. Water 2020, 12, x FOR PEER REVIEW 4 of 19

and Makhanda towns, where communities experienced severe water shortages and food security. Despite the persistence of these conditions and the inherent impacts, drought characteristics in the region have been understudied for years, with limitations being attributed to the complex influences Water 2020, 12, 1938 4 of 20 of weather systems (manifested from the midlatitudes and tropics), the proximity of the region to the Agulhas Currents, as well as the strong topographic gradients.

Figure 1. Eastern Cape province with the distribution of selected South African Weather Service Figure 1. Eastern Cape province with the distribution of selected South African Weather Service climate climate district centroids and other areal characteristics. district centroids and other areal characteristics. 3. Materials and Methods In recent years, the Eastern Cape has experienced a severe drought, accompanied by negative socio-economic3.1. Materials impacts across the province, with empirical evidence in Port Elizabeth, Graaﬀ-Reinet, and Makhanda towns, where communities experienced severe water shortages and food security. Data analyzed in the current study was based on the monthly district rainfall from 1968 to 2018. Despite the persistence of these conditions and the inherent impacts, drought characteristics in the A total of 22 rainfall climate districts distributed within the Eastern Cape province were selected and regionconsidered. have been These understudied districts form for years, part of with the limitations94 rainfall climate being attributeddistricts, which to the cover complex the whole inﬂuences of weathercountry, systems as delineated (manifested by the fromSouth the African midlatitudes Weather Service and tropics), (SAWS), the formally proximity known of theas the region South to the AgulhasAfrican Currents, Weather as Bureau well as 1972 the (e.g., strong SAWB topographic 1972). In general, gradients. these districts contain long-term monthly rainfall totals, from 1921 to present. The SAWS is responsible for updating the 94 rainfall districts and 3. Materialsthe process and is Methods performed monthly for detailed information on these rainfall climate districts the reader is referred to [38]. 3.1. Materials 3.2. Methods Data analyzed in the current study was based on the monthly district rainfall from 1968 to 2018. A total3.2.1. of 22Drought rainfall Definition climate districtsand Characterization distributed Using within the the Standardized Eastern Cape Precipitation province Index were selected and considered. These districts form part of the 94 rainfall climate districts, which cover the whole country, The SPI was developed with the basic understanding that a deficit in precipitation has cross-cut as delineated by the South African Weather Service (SAWS), formally known as the South African impacts on various types of water resources at different timescales, for example, ranging from soil Weather Bureau 1972 (e.g., SAWB 1972). In general, these districts contain long-term monthly rainfall moisture on a relatively short timescale and groundwater, reservoir storage, and streamflow, at long- totals,term from scale. 1921 Designed to present. to quantify The SAWS deficit isof responsibleprecipitation forat various updating timescales, the 94 the rainfall SPI can districts be used and as the processa monitoring is performed tool monthly for all the for detailedthree types information of drought, on namely these rainfallmeteorological, climate districtsagricultural, the readerand is referred to [38].

3.2. Methods

3.2.1. Drought Definition and Characterization Using the Standardized Precipitation Index The SPI was developed with the basic understanding that a deficit in precipitation has cross-cut impacts on various types of water resources at different timescales, for example, ranging from soil moisture on a relatively short timescale and groundwater, reservoir storage, and streamflow, Water 2020, 12, 1938 5 of 20 at long-term scale. Designed to quantify deficit of precipitation at various timescales, the SPI can be used as a monitoring tool for all the three types of drought, namely meteorological, agricultural, and hydrological. Computation of SPI requires precipitation as the solely hydrological input. The procedural approach described in [9] was followed in this study to calculate the SPI time series at 6 and 12 accumulation periods. In this case, the monthly precipitation series for each station were fitted to a gamma probability density function defined as per Equation (1),

1 α 1 x/β g(x) = x − e− (1) βαΓ(α) where α > 0 represents the shape parameter, β > 0 is a scalar parameter, x > 0 corresponds to the amount of precipitation and Γ(α) is the gamma function deﬁned as,

Z∞ a 1 y Γ(α) = y − e− dy (2) 0 where α and β parameters of the gamma distribution are based on the approximation of [39], as given in Equations (3) and (4)  r  1  4A α = 1 + 1 +  (3) 4A 3  P x ln(x) β = , with A = ln(x) (4) α − n where n is the number of observations. Integrating Equation (1) with respect to x leads to the computation of the cumulative probability function of the gamma distribution, expressed as in Equation (5),

Zx Zx Zx 1 α 1 x/β 1 a 1 1 G(x) = g(x)dx = x − e− dx = t − e− dt (5) βαΓ(α) Γ(a) 0 0 0

Transforming Equation (5) into the standard normal distribution yields to the SPI, a time series representing both positive (for wet conditions) and negative (for dry conditions) values. In order to characterize drought, we have used the run theory proposed by [40,41]. For this study, we have considered the drought duration and severity as two main drought characteristics derived using the algorithm summarized as follows:

(a) Define drought events/episodes as periods when the SPI values (SPI-6/-12) are negative (inclusive of zero) as reported in [42]. (b) Define drought epoch (DE) as the period when SPI values (SPI-6/-12) are consecutively zero and negative for six months. If during the six months, one month with positive SPI values occurs between the second and fourth month, the epochs before and after this intermittent occurrence of positive SPI value are combined to form a DE. (c) Drought duration (DD) is computed as the length of DE defined in (b), while drought severity (DS) is then computed as the integral of the SPI curve during the DE.

3.2.2. Multivariate Drought Analysis Using the Copulas In this study, the copula method was used to assess the relationship between DD and DS, calculated from the SPI-6 and SPI-12 time series. To model the joint probability distribution of drought Water 2020, 12, 1938 6 of 20 characteristics (duration and severity), it was assumed that for H(x,y), a joint distribution function, there exists a function expressed as in Equation (6),

H(x, y) = C(FX(x),FY(y)) (6)

In Equation (6), C is the copula, and FX(x) and FY(y) are the marginal distributions, thus the cumulative distribution functions of X and Y, respectively. Suppose U = FX(x) and V = FY(y), and given that X and Y are the marginal distributions, then the copula C can be deﬁned as ( ) 1( ) 1( ) C u, v = F FX− u ,FY− v

This allows the copula to be constructed as given by Equation (7). ( ( ) ( )) 1( ( )) 1( ( )) ( ) C FX x ,FY y = F FX− FX x ,FY− FY y = H x, y (7)

The marginal distributions for drought duration and severity were first identified, prior to the modelling of the joint probability distribution. This was achieved by fitting a number of probability distribution functions and assessing the best marginal distribution for the two drought features. The best fitted marginal distribution was identified based on the Akaike’s information criteria (AIC) and Bayesian information criteria (BIC). The tail dependency structure of drought duration and severity was assessed based on the Multivariate Copula Analysis Toolbox (MvCAT) reported by [43]. A total of 10 copula distribution families were assessed, these are summarized in Table1. After assessing the ranked scatter plot of drought characteristics, five copulas were selected for further analysis to identify the best copula. The correlation relationship between the drought characteristics was assessed by use of Kendall’s tau and Spearman’s correlation coefficients. Additionally, the best copula was identified based on the corresponding root mean square error (RMSE) and sorted based on the maximum likelihood. The best distribution function for the drought duration and severity was evaluated based on the best marginal distributions as well as the best.

Table 1. Summary of selected copula families adopted from [43]. The BB1 copula is derived from the combination of extreme cases of the Clayton and Gumbel copulas while BB5 is a 2-parameter extension of the Gumbel copula.

Name Mathematical Description Parameter Range References

1 1 2 2 R ∅− (u) R ∅− (v) 2θxy x y Gaussian 1 exp − − dxdyb θ[ 1, 1] [44] 2π √1 θ2 2(1 θ2) − −∞ −∞ − − 1/θ Clayton max u θ + v θ 1, 0 − θ[ 1, ) 0 [45] − − − − ∞ \ (exp( θu) 1)(exp( θv) 1) Frank 1 ln + − − − − θ 0 [44] θ 1 exp( θ) 1 R − − − \ h i1/θ Gumbel exp ( ln(u))θ + ( ln(v))θ θ[ 1, ) [44] − − − ∞ h i1/θ Joe 1 (1 u)θ + (1 v)θ (1 u)θ(1 v)θ θ[ 1, ) [44] − − − − − − − ∞ n θ θo 1/θ Galambos uv exp ( ln(u))− + ( ln(v))− − θ[0, ] [46] − − ∞ Cubic uv[1 + θ(u 1)(v 1)(2u 1)(2v 1)] θ[ 1, 2] [47] − − − − − ( 1/θ ) 1/θ1 θ2 θ2 2 − BB1 1 + u θ1 1 + v θ1 1 θ1(0, ), θ2(1, ) [48] − − − − ∞ ∞ ( ) 1/θ 1/θ1 θ1 θ1 θ1θ2 θ1θ2 2 BB5 exp ( ln(u)) + ( ln(v)) ( ln(u))− + ( ln(v))− − θ1[ 1, ) , θ2(0, ) [48] − − − − − − ∞ ∞ h i1/θ3 [ ] (1 θ1) (1 θ2) θ3 θ3 θ1, θ2 0, 1 , Tawn exp ln u − + ln v − ( θ1ln(u)) + ( θ2ln(v)) [46] − − − θ3[ 1, ) ∞ Water 2020, 12, 1938 7 of 20

3.2.3. Risk Analysis Framework-Based Return Periods The copula-based method was used to calculate the joint return periods for the considered drought characteristics. The following equations were used to calculate the joint drought return periods,

E(I ) E(I ) E(I ) T = d = d = d (8) DS P(D d or S s) 1 F (d, s) 1 C(F (d),F (s)) ≥ ≥ − DS − D S E(I ) E(I ) E(I ) T; = d = d = d (9) DS P(D d and S s) 1 F (d) F (s) 1 F (d) F (s) + C(F (d),F (s)) ≥ ≥ − D − S − D − S D S where FD(d) and FS(s) are cumulative marginal distribution functions of drought duration and severity, ( ) ; respectively; E Id is the expected time of inter-arrival time of drought events; and TDS and TDS are the joint return periods of drought duration and severity (see [49] for further details). For drought risks analysis, the annual return periods, T in years, (and the corresponding non-exceedance probability, q, see Equation (10)) were used. The multivariate drought risk analysis framework considers the duo drought variables: duration and severity thereby resulting to two forms of return periods akin to two non-exceedances commonly denoted as Tq;coop and Tq;dual as given in Equations (11) and (12), respectively. In essence, Equation (11) implies that both drought indices cooperate or collaborate as parts or as a whole, while Equation (12) signiﬁes that both drought indices are conjunctive or work together. 1 T = (10) 1 q − 1 1 1 Tq;coop = = (11) 1 C(q, q; Θ) { (1 q, 1 q; Θ) ≡ cooperative risk − ∗ − − where C(q,q;Theta) and C*(q0,q0;Theta) are the copula and the co-copula for the exceedance probability

q0 = 1 q, − respectively, that can be used to calculate the real return period of failure.

1 1 1 1 T = = = (12) q;dual ˆ ≡ complement o f dual protection 1 {e(q, q; Θ) {(q, q; Θ) {(q0, q0; Θ) − ~ ¯ In Equation (12), C(q, q; Θ), C(q, q; Θ) and Cˆ (q0, q0; Θ) are the dual of a copula function, the joint survival function and survival copula, respectively.

4. Results

4.1. Drought Characteristics The top panel of Figure2 depicts SPI-6 and SPI-12 time series for the selected climatic rainfall districts in the north-western part of the Eastern Cape province. Both SPI-6 and SPI-12 values show profound decadal drought variations. The two indices are ideally suited to describe the inherent total rainy seasons as well as the long-term drought conditions. Under the SPI-6 accumulation period, the three rainfall districts are characterized by, as reported in [13], levels 2 and 3 drought which is moderate-to-severe drought categories, thus ranging between 1.50 < SPI 1.00 and − ≤ − 2.00 < SPI 1.50, respectively. On the other hand, the SPI-12 depicts considerable decadal variability, − ≤ − wherein category 3 and 4 drought characteristics i.e., severe-to-extreme drought (SPI 2.00), ≤ − respectively. According to [16], such drought categories are likely to occur two to four times per century. Water 2020, 12, 1938 8 of 20

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FigureFigure 2. Standardized 2. Standardized Precipitation Precipitation Index Index (SPI (SPI time time series series for selected for selected climatic climatic rainfall districts rainfall in districts(A) in (A)north-western, north-western, (B) south, (B) south, and (C and) south-eastern (C) south-eastern regions of regions the Eastern of the Cape Eastern province. Cape Left province. panel: SPI- Left panel:6 SPI-6and right and panel: right panel:SPI-12. SPI-12.The blue The and blue red colo andrs red represent colors representwet and dr wety epochs, and dry respectively. epochs, respectively.

The south-easternThe south-eastern part ofpart the of study the areastudy (see area for (see example for example middle andmiddle bottom and panels)bottom experiencedpanels) moreexperienced frequent and more intense frequent drought and intense in the drought considered in the considered study period. study Inperiod. this regard,In this regard, the SPI-12 the in particular,SPI-12 illustrated in particular, an illustrated increase inan theincrease frequency in the frequency of drought of drought events atevents the respectiveat the respective rainfall rainfall districts. districts. Most of the drought occurrences in these rainfall districts were characterized by the level 4 Most of the drought occurrences in these rainfall districts were characterized by the level 4 drought drought category (e.g., SPI ≤ −2.00). Some of the drought occurrences between 1980/81 and 1990/91 category (e.g., SPI 2.00). Some of the drought occurrences between 1980/81 and 1990/91 were due were due to El≤ Nino, − a condition that brings very dry conditions along the south and south-eastern to Elpart Nino, of South a condition Africa [50,51]. that brings In general, very the dry results conditions demonstrate along that the the south south andand south-eastern south-eastern parts part of Southof Africa Eastern [50 Cape,51]. are In prone general, to prolonged the results and demonstrate frequent drought that theepisodes. south These and south-easternresults corroborate parts of Easternwith Cape a weekly are prone report to by prolonged the Department and frequentof Water and drought Sanitation episodes. (DWS; These31 January results 2019) corroborate which stated with a weeklythat report the Makhanda by the Department dam level, formerly of Water referred and Sanitation to as Grahamstown (DWS; 31 Januarywas at a critical 2019) which(approaching stated that the Makhandaday zero) level. dam A level, decrease formerly of 1.4% referred was observed to as Grahamstownwith a major drop was from at a57.4% critical to 56.0%. (approaching This has day zero)led level. to the A decrease intervention of 1.4% of the was national observed DWS with to partner a major with drop the from Makhanda 57.4% tolocal 56.0%. municipality This has to led to preclude a total disaster. the intervention of the national DWS to partner with the Makhanda local municipality to preclude a total disaster.

Water 2020, 12, 1938 9 of 20

InWater the 2020 current, 12, x FOR study, PEER REVIEW drought conditions were also assessed in terms of spatial distribution9 of 19 at 6- and 12-monthIn the current time steps.study, Indrought this regard, conditions the derivedwere also results assessed on in the terms contrasts of spatial of droughtdistribution episodes at 6- for SPI-6and and 12-month SPI-12 are time depicted steps. In in this Figure regard,3. Thethe derive pie chartsd results in Figureon the 3contrasts are the numericalof drought episodes proportions for of countsSPI-6 of drought and SPI-12 episodes are depicted derived in Figure from SPI-63. The andpie charts SPI-12 in time Figure series 3 are over the numerical the entire proportions study period. of It is evidentcounts from of drought the figure episodes that the derived percentage from SPI-6 of droughtand SPI-12 episodes time series identified over the entire using study SPI-6 period. was greater It thanis those evident identified from the basedfigure that on SPI-12the percentage (by 16.5%). of drought Furthermore, episodes identified results using show SPI-6 that was the greater study area exhibitsthan mean thosevalue identified of the based drought on SPI-12 episode (by of16.5%). 6/4 during Furthermore, the entire results study show period, that the as computedstudy area from SPI-(6exhibits/-12)-month mean accumulationvalue of the drought periods episode respectively. of 6/4 during It was the also entire determined study period, that as less computed than six from drought SPI-(6/-12)-month accumulation periods respectively. It was also determined that less than six episodes (computed from SPI-6) had a more likelihood of occurring (with a probability of ~0.8) while drought episodes (computed from SPI-6) had a more likelihood of occurring (with a probability of more than four drought episodes (derived from SPI-12) were more likely to occur (with a probability of ~0.8) while more than four drought episodes (derived from SPI-12) were more likely to occur (with a ~0.6)probability in the study of area.~0.6) in Figure the study4 depicts area. maximalFigure 4 depicts/extreme maximal/extreme drought periods drought based periods on SPI-6 based and on SPI-12 experiencedSPI-6 and during SPI-12 theexperienced considered during study the period.considered As study given period. in Figure As given4, the in study Figure area 4, the experienced study maximalarea droughtexperienced periods maximal that drought lasted for periods as little that as lasted 7/13 months for as little extending as 7/13 months to 33/48 extending months (for to 33/48 SPI-6 /12). The averagemonths (for maxima SPI-6/12). drought The average period maxima was determined drought period to be was approximately determined to 1.5 be and approximately 2.2 years based 1.5 on the SPI-6and 2.2 and years -12-accumulation based on the SPI-6 periods, and respectively.-12-accumulation Overall, periods, the resultsrespectively. illustrate Overall, that the the results study area was moreillustrate likely that (with the study a probability area was of more ~0.88) likely to experience (with a probability maximal of drought ~0.88) to conditions experience thatmaximal lastmore than 12drought months, conditions while droughtthat last more conditions than 12 lasting months over, while 24 drought months conditions had a 33% lasting likelihood over 24 of months occurrence. had a 33% likelihood of occurrence.

Figure 3. Proportion of occurrence of SPI-6 and SPI-12 drought episodes recorded across the Eastern Figure 3. Proportion of occurrence of SPI-6 and SPI-12 drought episodes recorded across the Eastern Cape province, South Africa. Cape province, South Africa.

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Figure 4. Spatial contrasts of maximal drought duration (in years) across the Eastern Cape province. Figure 4. Spatial contrasts of maximal drought duration (in years) across the Eastern Cape province.

4.2. Characteristics4.2. Characteristics of Marginal of Marginal Distributions Distributions Error! Reference source not found.Tables 2 and 3 give a summary of best marginal distributions Tables2 and3 give a summary of best marginal distributions fits to drought duration and severity fits to drought duration and severity computed from SPI-6 and SPI-12, across the selected rainfall computeddistricts, from respectively. SPI-6 and The SPI-12, correla acrosstion coefficient the selected was based rainfall on Kend districts,all’s tau respectively. and Spearman The tests correlation and coeffithesecient are was given based in oncolumns Kendall’s 4 and tau 5, andof each Spearman table. Based tests on and Table these 2 are(with given regard in to columns SPI-6), 4the and 5, of eachgeneralized table. Based pareto on Tableand 2gamma (with regarddistributions to SPI-6), dominate the generalized across the pareto region, and providing gamma distributionsthe best dominatedistribution across thefit for region, drought providing duration the tobest approximately distribution 28% fit forof th droughte study durationarea. In toaddition, approximately the 28% ofgeneralized the study pareto area. Infits addition, well to drought the generalized severity to pareto approximately fits well 23% to drought of the study severity area, to followed approximately by 23% ofRayleigh, the study inverse area, Gaussian, followed and by Weibull, Rayleigh, each inverse provid Gaussian,ing best fit and to about Weibull, 14% eachof the providing study area. best The fit to aboutSpearman’s 14% of the correlation study area. coefficient The Spearman’s was higher correlation across the region coeffi cientas compared was higher to Kendall’s across thetau. regionThe as comparedcoefficient to Kendall’s values ranged tau. Thebetween coeffi 0.4cient to 0.8 values for the ranged Spearman’s between test and 0.4 0.3 to 0.8to 0.6 for for the Kendall’s Spearman’s tau. test The calculated p-values (not shown) were statistically significant at 5% level across all the stations. and 0.3 to 0.6 for Kendall’s tau. The calculated p-values (not shown) were statistically significant at 5% Based on Table 3 (for SPI-12), the generalized pareto distribution provides the best fit for both leveldrought across all duration the stations. and severity across the selected rainfall districts. This distribution fits well to the Baseddrought on duration Table3 and(for severity, SPI-12), co thevering generalized 73% and pareto45% of distributionthe study area, provides respectively. the bestMost fit of for the both droughtrainfall duration districts and depict severity a strong across correlation the selected between rainfall the drought districts. duration This and distribution severity, as fits indicated well to the droughtfrom duration Spearman’s and severity,coefficient covering values. 73%These and values 45% ofranged the study (correlations) area, respectively. between 0.2 Most and of 0.7 the for rainfall districtsKendall’s depict tau a strongand 0.2 correlationto 0.8 for Spearman’s between test. the The drought p-values duration were statistically and severity, significant as indicated across all from Spearman’sbut two coerainfallfficient districts. values. These values ranged (correlations) between 0.2 and 0.7 for Kendall’s tau and 0.2 to 0.8 for Spearman’s test. The p-values were statistically significant across all but two rainfall districts.

Water 2020, 12, 1938 11 of 20

Table 2. Best probability distributions in modelling marginal drought duration and severity computed from Standardized Precipitation Index (SPI)-6 time series.

Fitted Distribution Correlation Coeﬃcient District No. Signiﬁcant at 5%? Spearman’s Duration Severity Kendall Rank Rank-Order 10 Generalized Pareto Rayleigh 0.566 0.745 Yes 11 Weibull Generalized Pareto 0.563 0.743 -do- 1 12 Generalized Pareto Rayleigh 0.514 0.667 -do- 13 Logistic Loglogistic 0.381 0.489 -do- 17 Rician Rayleigh 0.525 0.707 -do- 20 Weibull Generalized Pareto 0.394 0.539 -do- 21 Nakagami -do- 0.540 0.706 -do- 22 Loglogistic Inverse Gaussian 0.524 0.678 -do- 23 Generalized Pareto Nakagami 0.509 0.668 -do- 24 Gamma Weibull 0.278 0.364 -do- 27 Generalized Pareto Birnbaumsaunders 0.570 0.734 -do- 28 Normal Logistic 0.539 0.688 -do- 29 -do- -do- 0.539 0.688 -do- 39 Generalized Pareto Weibull 0.418 0.532 -do- 40 Birnbaumsaunders Gamma 0.499 0.640 -do- 41 Gamma Birnbaumsaunders 0.510 0.653 -do- 42 Birnbaumsaunders Inverse Gaussian 0.558 0.718 -do- 43 Rician Generalized Pareto 0.351 0.463 -do- 55 Loglogistic Inverse Gaussian 0.519 0.643 -do- 56 Gamma Generalized Pareto 0.498 0.645 -do- 57 Generalized Pareto Weibull 0.602 0.737 -do- 58 Gamma Nakagami 0.441 0.587 -do- 1 -do-: denotes “as above”.

Table 3. Best probability distributions in modelling marginal drought duration and severity computed from SPI-12 time series.

Fitted Distribution Correlation Coeﬃcient District No. Spearman’s Signiﬁcant at 5%? Duration Severity Kendall Rank Rank 10 Generalized Pareto Rayleigh 0.512 0.661 Yes 11 -do- Generalized Pareto 0.671 0.813 -do- 12 -do- -do- 0.674 0.831 -do- 13 -do- -do- 0.606 0.770 -do- 17 -do- -do- 0.581 0.759 -do- 20 -do- Birnbaumsaunders 0.442 0.577 -do- 21 -do- Inverse Gaussian 0.576 0.736 -do- 22 -do- Generalized Pareto 0.551 0.704 -do- 23 Rician -do- 0.473 0.629 -do- 24 Generalized Pareto Gamma 0.151 0.199 No 27 -do- -do- 0.151 0.199 -do- 28 -do- Generalized Pareto 0.634 0.782 Yes 29 Loglogistic Weibull 0.418 0.542 -do- Water 2020, 12, 1938 12 of 20

Table 3. Cont.

Fitted Distribution Correlation Coeﬃcient District No. Spearman’s Signiﬁcant at 5%? Duration Severity Kendall Rank Rank 39 Generalized Pareto Generalized Pareto 0.539 0.685 -do- 40 -do- Gamma 0.507 0.693 -do- 41 -do- Rayleigh 0.492 0.644 -do- 42 Rician Nakagami 0.627 0.783 -do- 43 Gamma Generalized Pareto 0.403 0.500 -do- 55 Weibull -do- 0.652 0.801 -do- 56 Generalized Pareto Gamma 0.495 0.635 -do- 57 -do- Extreme value 0.619 0.790 -do- 58 Rician Inverse Gaussian 0.334 0.409 -do-

4.3. Copula Joint Probability Distributions Of the 10 copula families analyzed, five copulas were found to be suitable and therefore selected for the assessment of joint drought characteristics, thus the joint drought duration and severity features. A further analysis lead to the selection of two copulas for the analysis of joint drought characteristics, see results presented in Table4. The selection of the two copulas was based on assessing the RMSE values across the copulas and the rainfall districts. The RMSE values were ranked and based on max likelihood, the copulas were sorted from the least RMSE value to the highest value. The selected copulas are a combination of single, two and three parameters, with the computed values depicted in the tables. The best copula for each rainfall district is indicated in red, and this is Tawn copula for both SPI-6 and SPI-12 drought duration and severity features, followed by BB1 for SPI-6 and Joe for SPI-12. The Tawn copula had three parameters, whereas Joe and BB1 contained one and two parameters, respectively, see the estimated values in Table4. The fact that Tawn copula predominates in both cases suggests that the three-parameter copulas were more suitable to assess the dependency of drought duration and severity. Figures5 and6 depict plots of dependence structure between drought duration and severity represented at different probability space levels at a selected SAWS climate district. These were constructed from the best marginal distributions and best copula (e.g., Tawn and BB1 for SPI-6 and Tawn and Joe for SPI-12). In both cases, there is a tendency of asymmetric and skewness dependence structure of the drought features. There exist slight differences in the isolines produced by Tawn and BB1 copulas for SPI-6, and Tawn and Joe copulas for SPI-12, particularly between the 10th and 20th percentiles. Such differences could be attributed to tail dependence of the copulas, e.g., the Tawn copula has no tail dependence whereas BB1 and Joe have tail dependence coefficients. Nevertheless, the probability isolines derived from all the copulas are notably skewed to the left, suggesting a low probability of drought duration and high probability of drought severity.

Table 4. Selected copula joint probability distributions and their characteristics. Red indicates best copula for the climatic rainfall district. The RMSE denotes the Root Mean Square Error.

SPI-6 SPI-12 Rainfall Dist. No. Tawn BB1 Tawn Joe

RMSE θ1 RMSE θ1 RMSE θ1 RMSE θ1 10 0.252 0.883 0.214 2.282 0.301 0.999 0.298 4.821 11 0.329 0.759 0.288 3.255 0.331 0.671 0.356 4.390 12 0.273 0.773 0.278 0.207 0.291 0.997 0.280 6.620 Water 2020, 12, 1938 13 of 20

Table 4. Cont.

SPI-6 SPI-12 Rainfall Dist. No. Tawn BB1 Tawn Joe

RMSE θ1 RMSE θ1 RMSE θ1 RMSE θ1 13 0.375 0.758 0.381 0.213 0.515 0.942 0.498 6.228 17 0.298 0.761 0.313 0.056 0.331 0.997 0.325 13.730 20 0.374 0.779 0.352 3.473 0.399 0.746 0.398 5.287 21 0.358 1.000 0.348 9.936 0.377 0.998 0.358 7.129 22 0.438 0.834 0.450 0.000 0.287 0.772 0.295 5.008 23 0.343 0.816 0.350 0.010 0.324 0.766 0.359 6.752 24 0.393 0.486 0.476 0.006 0.817 0.486 0.832 2.688 27 0.676 0.991 0.675 0.000 0.817 0.505 0.832 2.680 28 0.345 0.862 0.354 0.000 0.360 0.919 0.360 9.953 29 0.345 0.863 0.354 0.000 0.449 0.787 0.458 6.559 39 0.697 0.719 0.715 0.000 0.628 0.994 0.626 17.762 40 0.364 0.903 0.366 3.433 0.420 0.956 0.385 0.000 41 0.321 0.802 0.321 0.263 0.341 0.879 0.334 3.296 42 0.331 0.906 0.332 0.000 0.309 0.805 0.323 5.926 43 0.386 0.763 0.376 1.759 0.386 0.871 0.410 6.529 55 0.308 0.826 0.323 0.018 0.261 0.999 0.269 26.656 56 0.313 0.818 0.310 0.908 0.273 0.843 0.300 4.126 57 0.306 0.849 0.297 2.982 0.407 0.852 0.422 11.982 58 0.356 0.712 0.352 1.786 0.293 0.686 0.327 4.791

4.4. Risk Assessment Based on the Join Drought Return Periods Given that drought is a multivariate event, it is, therefore, best to fully account the inherent risks using a combination of two or more drought monitoring indicators. In the present study, the drought risks were assessed using the joint return periods derived from the Joe (and Gaussian, not shown) copula functions. In this case, the copula functions were calculated from the marginal distributions ﬁtted by the two drought monitoring indicators i.e., drought duration and severity e.g., the SPI-6. The spatial distribution of joint return periods for the dual and cooperative cases (see Equations (10) and (11)) alongside the 2-, 5-, 10-, 20- and 100-year univariate return period are given in Figures7 and8, respectively. As depicted in Figure7, the multivariate return period for the dual case was always longer across all the selected univariate return periods manifesting an overestimate of the drought risks. The dual (conjunctive) case contrasts cooperative multivariate case as shown in Figure8, a consequence of the formulations given in Equations (10) and (11). In terms of drought risk analysis, the study area exhibited a noticeable north-west to south-east gradient with Buﬀalo City, Tambo and Alfred Zoo regions determined to have higher/lower risks in terms of the conjunctive/cooperative multivariate drought risk (copula) probability index. Water 2020, 12, x FOR PEER REVIEW 13 of 19

Figures 5 and 6 depict plots of dependence structure between drought duration and severity represented at different probability space levels at a selected SAWS climate district. These were constructed from the best marginal distributions and best copula (e.g., Tawn and BB1 for SPI-6 and Tawn and Joe for SPI-12). In both cases, there is a tendency of asymmetric and skewness dependence structure of the drought features. There exist slight differences in the isolines produced by Tawn and BB1 copulas for SPI-6, and Tawn and Joe copulas for SPI-12, particularly between the 10th and 20th percentiles. Such differences could be attributed to tail dependence of the copulas, e.g., the Tawn copula has no tail dependence whereas BB1 and Joe have tail dependence coefficients. Nevertheless, the probability isolines derived from all the copulas are notably skewed to the left, suggesting a low Water 2020, 12, 1938 14 of 20 probability of drought duration and high probability of drought severity.

FigureFigure 5. A 5. typicalA typical dependence dependence structure structure ofof droughtdrought duration duration and and severity severity based based on SPI-6 on SPI-6 time series.time series. In theIn the figure ﬁgure (a ()a corresponds) corresponds to to BB1 andand ( b(b)) corresponds corresponds to Tawnto Tawn copulas. copulas.

Water 2020, 12, 1938 15 of 20 Water 2020, 12, x FOR PEER REVIEW 14 of 19

FigureFigure 6. 6.AA dependence dependence structure of of drought drought duration duration and and severity severity based based on SPI-12 on SPI-12 time series. time series. In the In the ﬁgure, (a) corresponds to BB1 and (b) corresponds to Joe copulas. figure, (a) corresponds to BB1 and (b) corresponds to Joe copulas.

Water 2020, 12, x FOR PEER REVIEW 15 of 19

longer across all the selected univariate return periods manifesting an overestimate of the drought risks. The dual (conjunctive) case contrasts cooperative multivariate case as shown in Figure 8, a consequence of the formulations given in Equations (10) and (11). In terms of drought risk analysis, the study area exhibited a noticeable north-west to south-east gradient with Buffalo City, Tambo and Water 2020Alfred, 12 ,Zoo 1938 regions determined to have higher/lower risks in terms of the conjunctive/cooperative16 of 20 multivariate drought risk (copula) probability index.

FigureFigure 7. Spatial 7. Spatial distribution distribution ofof conjunctiveconjunctive return return period period corresponding corresponding to (a) to2-year, (a) 2-year, (b) 5-year, (b) (c 5-year,) 10-year, (d) 20-year, and (e) 100-year univariate risks of return periods based on the Joe Copula. (Waterc) 10-year, 2020, 12 (, dx )FOR 20-year, PEER REVIEW and (e) 100-year univariate risks of return periods based on the Joe Copula.16 of 19

FigureFigure 8. Spatial 8. Spatial distribution distribution ofof cooperative return return period period corresponding corresponding to (a) to2-year, (a) 2-year,(b) 5-year, (b) (c 5-year,) 10-year, (d) 20-year, and (e) 100-year univariate risks of return periods based on the Joe Copula. (c) 10-year, (d) 20-year, and (e) 100-year univariate risks of return periods based on the Joe Copula. 5. Discussion and Conclusions South Africa is generally known to be a water-stressed country because the available water resources do not meet the ever-increasing demand for e.g., domestic, industrial, and agricultural use. This situation is often exacerbated by the prolonged drought conditions albeit the intermittent flooding episodes. For instance, between 2014 and 2019, South Africa experienced persistent drought conditions which led to the declaration of drought as a national disaster (in March 2018) and many cities across the country such as Cape Town (Western Cape province), Port Elizabeth (Eastern Cape province), and Phalaborwa (Limpopo province) nearly approached “day-zero” state. Studies, such as the current one, are therefore invaluable for drought monitoring, drought risk assessment, preparedness, and mitigation with applications in water resource management. It is for this purpose that, the present study has been undertaken to characterize drought conditions across the Eastern Cape province based on a multivariate risk assessment framework using a combination of drought duration and drought severity indicators. The literature on drought analysis over South Africa is generally abound, a comprehensive assessment of drought risks at provincial scale and from the viewpoint of the drought risk probability index [52,53] across South Africa remains in-exhaustive and yet nascent. The present analysis results point to the following conclusions; (a) Five to eight drought episodes (with an average of 60–80% likelihood of occurrence) were experienced in the study area over the last five decades. (b) The Eastern Cape province is more likely to experience maxima drought conditions that last more than 12 months while drought conditions lasting over 24 months have a 33% likelihood of occurrence. (c) Five copulas families, from the Archimedean and Elliptical families, are duly suited to represent the multivariate characteristics of drought conditions in the study area. The spatial signature of the return periods from the five copulas was found to be

Water 2020, 12, 1938 17 of 20

5. Discussion and Conclusions South Africa is generally known to be a water-stressed country because the available water resources do not meet the ever-increasing demand for e.g., domestic, industrial, and agricultural use. This situation is often exacerbated by the prolonged drought conditions albeit the intermittent ﬂooding episodes. For instance, between 2014 and 2019, South Africa experienced persistent drought conditions which led to the declaration of drought as a national disaster (in March 2018) and many cities across the country such as Cape Town (Western Cape province), Port Elizabeth (Eastern Cape province), and Phalaborwa (Limpopo province) nearly approached “day-zero” state. Studies, such as the current one, are therefore invaluable for drought monitoring, drought risk assessment, preparedness, and mitigation with applications in water resource management. It is for this purpose that, the present study has been undertaken to characterize drought conditions across the Eastern Cape province based on a multivariate risk assessment framework using a combination of drought duration and drought severity indicators. The literature on drought analysis over South Africa is generally abound, a comprehensive assessment of drought risks at provincial scale and from the viewpoint of the drought risk probability index [52,53] across South Africa remains in-exhaustive and yet nascent. The present analysis results point to the following conclusions;

(a) Five to eight drought episodes (with an average of 60–80% likelihood of occurrence) were experienced in the study area over the last five decades. (b) The Eastern Cape province is more likely to experience maxima drought conditions that last more than 12 months while drought conditions lasting over 24 months have a 33% likelihood of occurrence. (c) Five copulas families, from the Archimedean and Elliptical families, are duly suited to represent the multivariate characteristics of drought conditions in the study area. The spatial signature of the return periods from the five copulas was found to be comparable. These copula families are therefore ideal for drought risk quantification in the study area. (d) The conjunctive/cooperative multivariate drought risk (copula) probability index results illustrate that the area exhibits a noticeable north-west to south-east gradient with Buffalo City, Tambo and Alfred Zoo regions determined to have higher/lower risks.

This study, therefore, provides an invaluable, robust, and representative copula framework for a holistic drought risk analysis at provincial scale. It is therefore recommended that the methodology be extended for drought analysis across diﬀerent provinces of South Africa and the southern African region, in support for drought preparedness, monitoring, and prediction.

Author Contributions: C.M.B. conceptualized, crafted the original draft, and finalized the manuscript. J.O.B. conceptualized, undertook the data analysis, discussed the results, and edited and approved the manuscript. J.P.d.W. prepared the study area map and edited the manuscript. A.M.A. generated the maps, contributed towards the discussion of the results, and edited the manuscript. N.N.Z. contributed towards the discussion of the results and edited the manuscript. K.P.N. discussed the results and edited the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by funding received from the Water Research Commission, South Africa and for this, the authors are grateful. Acknowledgments: The authors wish to thank the three anonymous reviewers for providing extensive comments and suggestions that lead to the improvement of the manuscript quality. Conflicts of Interest: The authors declare no conflict of interest.

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